A conventional approach in the signal processing applications listed above is to use a time domain approach, where a filterbank is not used, and a single adaptive filter acts on the entire frequency band of interest. This single time domain filter is typically required to be very long, especially when applied to acoustic echo cancellation. Computational requirements are a concern because longer filters require exponentially increasingly more processing power (i.e., doubling the filter length increases the processing requirements by more than two). A longer filter typically requires more iterations by its adaptive controlling algorithm to converge to its desired state. In the case of an adaptive noise cancellation algorithm, slow convergence hampers the ability of the system to quickly reduce noise upon activation and to track changes in the noise environment.
In summary, the problems with time domain adaptive signal processing are: 1) Long filters are required—cannot interleave the update of multiple filters. 2) Slower filter convergence due to longer filter length, 3) Performance problems in the presence of coloured noise, and 4) Inability to set varying algorithm parameters for individual frequency bands.
Solutions to problems in time domain adaptive signal processing arising from coloured noise and a long filter are limited. A long filter is often a requirement that is dictated by the particular application, and shortening it would degrade performance. In cases when it is allowable, white noise can be inserted into the signal path to allow the filter to adapt quicker.
Slow convergence is usually dealt with by choosing algorithm parameters that result in fast convergence while still guaranteeing filter stability. In the Least Mean Squares (LMS) algorithm this is done by increasing the step-size parameter (mu). However, this approach causes considerable distortion in the processed output signal due to the larger fluctuations of the adaptive filter resulting from a high mu value.
A method used to increase computational speed in time domain signal processing is to perform operations in the Fourier transform domain (see J. J. Shynk, “Frequency Domain and Multirate Adaptive Filtering”, IEEE Signal Processing Magazine, vol. 9, no. 1, pp. 15–37 January 1992). A section of the signal is transformed, operated on, then undergoes an inverse transformation. Methods are well known for performing specific operations in the transform domain that directly correspond to linear convolution (a common operation) in the time domain, but require less processing time. The added requirement of having to calculate the Fourier transform and inverse Fourier transform is offset when the signal can be transformed in blocks that are sufficiently large.